If $f (x + a y, x - a y) = a x y$ then $f (x, y)$ is equal to :

\frac{x^{2} - y^{2}}{4}

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\frac{x^{2} + y^{2}}{4}

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4 x y

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none

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Solution

The correct option is B $\frac{x^{2} - y^{2}}{4}$
Given, $f (x + a y, x - a y) = a x y$
Let $X = x + a y, Y = x - a y$
Solving these we get $x = \frac{X + Y}{2}$ and $y = \frac{X - Y}{2 a}$
Thus $f (X, Y) = \frac{X^{2} - Y^{2}}{4}$
Now replace $X \to x$ and $Y \to y$
$∴ f (x, y) = \frac{x^{2} - y^{2}}{4}$

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